Apple ID payment

This direct quote from Apple may provide better clarity as to how exactly the payment system works:

From this support article --> How apps, content, and subscriptions from Apple are billed - Apple Support

"Which payment method is charged?

When you buy items or pay for subscriptions, Apple automatically attempts to charge your payment methods in this order:

1. Apple attempts to use your Apple ID balance to cover the full amount. [This is your "default payment method".]

2. If you don't have Apple ID balance, you don't have enough to cover the full amount, or an item cannot be paid for with Apple ID balance, Apple attempts to charge your primary payment method (typically a credit or debit card, but varies according to country). The primary payment method is at the top of the list in Settings on your iPhone. You can control which payment method is at the top of the list [See: Add a payment method to your Apple ID - Apple Support.]

3. If there's a problem with the primary payment method, or if the primary payment method can't be charged for the full remaining amount, Apple attempts to charge your other payment methods in order from top to bottom. If none of them can be charged, Apple may try again."

You can have multiple cards entered using the method in this article: Add a payment method to your Apple ID - Apple Support

If it isn't working on one device, try using a different device if possible. You can also try restarting your device since that sometimes clears up odd issues. If you still can't get it to work, contact Apple. Refer to this document for ways to contact Apple --> Choose your country or region - Official Apple Support

Select your country (also look for "other" regions), then a product. If you don't see one that handles your issue then keep experimenting with selections until you reach one that gets you a chat session or a telephone call and get the representative to redirect you.

or:

Contact Apple for support and service by telephone --> Contact Apple for support and service - Apple Support

I’m simply unable to add a second form of payment of any kind.

If you mean 2 CC, you have to set one CC as your default payment method that will be charge by Apple.

As posted:

You can have a credit card / debit card on file and an Apple Account balance.

I have linked a CC to my Apple ID and also use Apple ID balance.

You can have a credit card / debit card on file and an Apple Account balance.

Your default payment method will be used first by Apple when you need to make a payment.  

If you have set your Apple Account balance as your default payment method and you have a CC / debit card, you pay from your default payment method.

If you have 2 CC, and you have set one CC as your default payment method that will be charge by Apple.